Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Abstract

For every Banach space X with a Schauder basis consider the Banach algebra RI ⊕ K diag (X) of all diagonal operators that are of the form λI + K. We prove that RI ⊕ K diag (X) is a Calkin algbra i.e., there exists a Banach space Y X so that the Calkin algebra of Y X is isomorphic as a Banach algebra to RI ⊕ K diag (X). Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces Jp and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.where the supremum is taken over all disjoint collections of intervals of N, is finite.The spaces J p are quasi-reflexive of order one. It was for observed by A. D. Andrew and W. L. Green in [AG] that J p , after appropriate renorming, becomes a non-unital Banach algebra when endowed with coordinate-wise multiplication with respect to the basis e 1 , e 2 − e 1 , e 3 − e 2 , . . .. We denote the unitization of James space by Re ω ⊕ J p , for 1 < p < ∞.Theorem III. The spaces Re ω ⊕ J p , 1 < p < ∞ are Calkin algebras.S. F. Bellenot, R. G. Haydon, and E. Odell in [BHO] extended the definition of James, based on the unit vector basis of ℓ p , to an arbitrary space X with an unconditional basis to define the space J(X), the "jamesification" of X. The space J(X) is quasi-reflexive of order one whenever X is reflexive. As it so happens this space is a non-unital Banach algebra as well the unitization of which we denote by Re ω ⊕ J(X). Furthermore, a special subspace of J * (X) which we denote by J * (X), and coincides with J(X) * when X does not contain ℓ 1 , is a separable unital Banach algebra.Theorem IV. For any space X with a normalized unconditional basis the spaces Re ω ⊕ J(X) and J * (X) are Calkin algebras.Next we turn our attention to spaces with unconditional bases endowed with coordinate-wise multiplication. We study these spaces themselves as Banach algebras. We do not prove that they are Calkin algebras but they always embed in them as complemented ideals. In what follows "⊕" denotes the direct sum of Banach spaces.Theorem V. Let A be Banach space with a normalized unconditional basis endowed with coordinate-wise multiplication and B be a Banach algebra. In the cases described in the list bellow there exists a Banach space Y so that the Calkin algebra Cal(Y) contains an idealà isomorphic as a Banach algebra to A and a subalgebrã B isomorphic as a Banach algebra to B so that Cal(Y) =à ⊕B.(a) The space B is C(ω), the space of convergent scalar sequences with pointwise multiplication.

Separable Spaces of Continuous Functions as Calkin Algebras

“…From a Banach-space-theoretic perspective, the construction of a reflexive Calkin algebra seems particularly intriguing and challenging. In [28] a quasireflexive Cal (X) was found. Although this may seem very close, the underlying space X has an infinite dimensional Schauder decomposition.…”

“…Iterating this process yields, for every countable compactum K, a C(K) Calkin algebra. A similar method was used by Puglisi, Tolias, and the author in [28] to construct a variety of Calkin algebras, e.g., quasi-reflexive and hereditarily indecomposable ones. All underlying Banach spaces mentioned in this paragraph may be viewed as composite Argyros-Haydon spaces.…”

“…It is worth mentioning that, if viewed simply as Banach spaces, there exists a large variety of Calkin algebras. In [28] the author, Puglisi, and Tolias proved that there exist HI Calkin algebras and that every non-reflexive space with an unconditional basis is a Calkin algebra, albeit with a nonstandard multiplication.…”

Banach spaces for which the space of operators has 2𝔠closed ideals

“…Then, as observed in [12], the closed ideals in L( ) arise from preimages of closed ideals in . Examples of separable Banach spaces for which L( )/K( ) has exactly continuum many closed ideals were constructed, for instance, in [30], [18] and [17], as explained in [9]. An example of a space for which the number of closed ideals is infinite but countable seems to be missing.…”

“…The first author was supported by grant 353293 from the Simons Foundation, and the second author's research was supported by NSF grant DMS-1764343. We want to thank the referee for improving our paper and for pointing out to us the relevant references [15], [17] and [9].…”